What size chain for Shimano CS-M7771-10 (11-36T) cassette?

There’s no need to worry about specified width. The bottom line is that any 10 speed chain from a reputable manufacturer (Shimano, SRAM, KMC, Wipperman) will work fine with a Shimano cassette.

All 10 speed chains should be compatible with 10s cassettes. The nominal widths I'm seeing are all about 5.88mm. Specifically, this is the distance between the inner faces of the inner plates, as diagrammed by Bike Gremlin.

There's one possible exception to the above: early Campagnolo 10s chains, released in the early 2000s when Shimano was still on 9s, are 6.2mm wide. Campy later changed to 5.9mm. In a pinch, I think older Campy chains would do on a Shimano 10s cassette, but I suspect they would be noisier and might not shift as well. The 10s Campy sprockets are slightly thicker than the Shimano 10s, and the average cog-cog distance is a bit bigger on Campy. Of course, I doubt you are getting your hands on an older 10s Campagnolo chain. There probably is some variation in mean chain width among manufacturers, but it's not important in practice.

As an aside, in theory, manufacturers may design the shaping of the links on their chains to interface with the shift ramps on the cassette sprockets. From people I've spoken to, received wisdom seems to be that you should ideally match the manufacturers for cassettes and chains, or else use a third party chain like KMC or Wipperman. In practice, the differences should mostly be ignorable. From a recent Velonews podcast on chains, the Shimano rep seemed to imply that this issue applied more strongly to Shimano's 11s and subsequent chains, so it may be less prominent on earlier stuff. I've heard friends use KMC chains on Shimano 11s drivetrains, however.